Embedding and a priori wavelet-adaptivity for Dirichlet problems (Q2707091)

Let \(\Omega\subset R^d\) be a bounded domain with Lipschitz continuous boundary embedded in a rectangular domain \(\Theta\). The Dirichlet boundary value problem in \(\Omega\) is extended to a periodic boundary value problem over \(\Theta\). For the corresponding variational problem (called the fictitious domain formulation), a Galerkin discretization is applied with subspaces generated by integer translations and binary dilations of a single biorthogonal compactly supported scaling function. Such a domain embedding method was proposed and analyzed in the author's previous paper [RAIRO, Modélisation Math. Anal. Numér. 32, No. 4, 405-431 (1998; Zbl 0913.65099)].NEWLINENEWLINENEWLINENow, in order to improve accuracy, the author proposes an a priori adaptive strategy using a finer discretization near the boundary of \(\Omega\). This is done by a modification of approximation subspaces via selecting suitable wavelet subspaces. The theoretical result is illustrated by numerical experiments.